Stochastic domination and Markovian couplings

Markovian couplings staying in arbitrary subsets of the state space

“…The results in this paper extend those of [12] in two significative aspects. On the one hand, the state space of the chains need not be the same, which is a basic assumption in the construction of order-preserving couplings as considered in [12].…”

“…The results in this paper extend those of [12] in two significative aspects. On the one hand, the state space of the chains need not be the same, which is a basic assumption in the construction of order-preserving couplings as considered in [12]. On the other hand, when the state spaces coincide, the set K where the coupling is required to stay is completely arbitrary (in particular, it need not be induced by a pre-ordering on E).…”

“…This is the object of Theorem 2, which shows that the points of E having some properties related with the structure of K need not be duplicated for the construction of the coupling. In fact, when K is defined by a pre-ordering on E, the construction of the coupling given in Theorem 2 coincides with the case of order-preserving couplings on partially ordered spaces [12].…”

“…Kamae et al [6] study the relationship between stochastic comparison on partially ordered Polish spaces and couplings, showing that (X t ) is stochastically dominated by (Y t ) if and only if there exists an order-preserving coupling of (X t ) and (Y t ), that is, a coupling (Z t ) such that P z (Z t ∈ K) = 1 for all t > 0 if z ∈ K, where K = {z = (x, y) : x ≤ y} and P z (Z t ∈ K) denotes the probability of the event {Z t ∈ K} when starting from z. In [12] it is shown that if (X t ) and (Y t ) are continuous-time Markov chains on a partially ordered countable set E and (X t ) is stochastically dominated by (Y t ), then the order-preserving coupling (Z t ) can be chosen to be a Markov chain (i.e. a Markovian coupling); also, a way to construct the coupling is given.…”

“…In this paper, we completely solve the aforementioned problem for general Markov chains (Theorem 1) without any hypotheses on the rates or the structure of K. Our approach relies on the definition of a pre-ordering on E ∪ F based on K and the use of the ideas and results in [12]. This will allow us to get necessary and sufficient conditions on the rates for the existence of a K-coupling.…”